Question

Cake Area ( in.2
in.
2
) Cake Pan Radius
(in.)
0 0
25 2.8
40 3.54
55 4.15
70 4.69
The table shows the radius of a cake pan as a function of the cake’s base area. Use technology to create a square root function that models this data. Round values to the nearest hundredth.

Answer 1

To create a square root function that models the given data, we will need to define a mathematical relationship between the cake area and the cake pan radius. Given the nature of the data, a square root function can be expressed in the form:

\[
r(A) = k \sqrt{A}
\]

where:
- \( r \) is the radius of the cake pan,
- \( A \) is the area of the cake,
- \( k \) is a constant to be determined.

After rearranging the equation, we have:

\[
\frac{r(A)}{\sqrt{A}} = k
\]

We can calculate \( k \) for each point from the data set to find a suitable model. Here’s the data provided:

| Cake Area (in\(^2\)) | Cake Pan Radius (in.) |
|-----------------------|------------------------|
| 0 | 0 |
| 25 | 2.8 |
| 40 | 3.54 |
| 55 | 4.15 |
| 70 | 4.69 |

### Step 1: Calculate \( k \)

Let's calculate \( k \) for the non-zero areas:

1. For \( A = 25 \):
\[
k = \frac{2.8}{\sqrt{25}} = \frac{2.8}{5} = 0.56
\]

2. For \( A = 40 \):
\[
k = \frac{3.54}{\sqrt{40}} \approx \frac{3.54}{6.32} \approx 0.56
\]

3. For \( A = 55 \):
\[
k = \frac{4.15}{\sqrt{55}} \approx \frac{4.15}{7.42} \approx 0.56
\]

4. For \( A = 70 \):
\[
k = \frac{4.69}{\sqrt{70}} \approx \frac{4.69}{8.37} \approx 0.56
\]

### Step 2: Confirm the value of \( k \)

Across the measurements, it appears that \( k \approx 0.56 \) consistently. Thus, we can say that the square root function that models this data is:

\[
r(A) \approx 0.56 \sqrt{A}
\]

### Final Model:

To summarize, the final function that models the radius of the cake pan as a function of the cake's area is:

\[
r(A) \approx 0.56 \sqrt{A}
\]

### Rounding:

Since the problem requests rounding to the nearest hundredth, we find:

\[
k \approx 0.56
\]

Thus, the rounded model function remains:

\[
r(A) \approx 0.56 \sqrt{A}
\]

This function can now be used to estimate the cake pan radius for any cake area within the observed range.